The k-Rainbow Bondage Number of a Digraph

Let D = (V,A) be a finite and simple digraph. A k-rainbow dominating function (kRDF) of a digraph D is a function f from the vertex set V to the set of all subsets of the set {1, 2, . . . , k} such that for any vertex v ∈ V with f(v) = Ø the condition ∪ f(u) = {1, 2, . . . , k} is fulfilled, where N (v) is the set of in-neighbors of v. The weight of a kRDF f is the value w(f) = ∑ |f(v)|. The k-rainbow domination number of a digraph D, denoted by γ (D), is the minimum weight of a kRDF of D. The k-rainbow bondage number b (D) of a digraph D with maximum in-degree at least two, is the minimum cardinality of all sets A′ ⊆ A for which γ (D−A′) > γ (D). In this paper, we establish some bounds for the k-rainbow bondage number and determine the k-rainbow bondage number of several classes of digraphs.